Approximating Minimum-Area Rectangular and Convex Containers for Packing Convex Polygons

MINIMUM AREA CONVEX PACKING OF TWO CONVEX POLYGONS

International Journal of Computational Geometry & Applications ◽

10.1142/s0218195906001926 ◽

2006 ◽

Vol 16 (01) ◽

pp. 41-74 ◽

Cited By ~ 6

Author(s):

KAI TANG ◽

CHARLIE C. L. WANG ◽

DANNY Z. CHEN

Keyword(s):

Convex Hull ◽

Complexity Analysis ◽

Search Space ◽

Deterministic Algorithm ◽

Algebraic Structures ◽

Convex Polygons ◽

Minimum Area ◽

Area Function ◽

Algebraic Properties ◽

Algorithmic Design

Given two convex polygons P and Q in the plane that are free to translate and rotate, a convex packing of them is the convex hull of a placement of P and a placement of Q whose interiors do not intersect. A minimum area convex packing of P and Q is one whose area is minimized. The problem of designing a deterministic algorithm for finding a minimum area convex packing of two convex polygons has remained open. We address this problem by first studying the contact configurations between P and Q and their algebraic structures. Crucial geometric and algebraic properties on the area function are then derived and analyzed which enable us to successfully discretize the search space. This discretization, together with a delicate algorithmic design and careful complexity analysis, allows us to develop an efficient O((n + m)nm) time deterministic algorithm for finding a true minimum area convex packing of P and Q, where n and m are the numbers of vertices of P and Q, respectively.

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Study on the Minimum Area of Emergency Refuge Room in the Single-Hole Bidirectional Highway Tunnel

CICTP 2020 ◽

10.1061/9780784483053.390 ◽

2020 ◽

Author(s):

Jiangbi Hu ◽

Chike Yuan ◽

Ronghua Wang ◽

Xu Wang

Keyword(s):

Minimum Area ◽

Single Hole ◽

Highway Tunnel

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Certain Two-Dimensional Local Limit Theorems with Applications to the Statistics of Convex Polygons

SSRN Electronic Journal ◽

10.2139/ssrn.3536633 ◽

2020 ◽

Author(s):

Sakhavet Zarbaliev

Keyword(s):

Limit Theorems ◽

Local Limit ◽

Two Dimensional ◽

Convex Polygons ◽

Local Limit Theorems

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Convex-Straight-Skeleton Voronoi Diagrams for Segments and Convex Polygons

Algorithmica ◽

10.1007/s00453-021-00824-9 ◽

2021 ◽

Author(s):

Gill Barequet ◽

Minati De ◽

Michael T. Goodrich

Keyword(s):

Voronoi Diagrams ◽

Convex Polygons

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Foraging range and habitat use by Cape Vulture Gyps coprotheres from the Msikaba colony, Eastern Cape province, South Africa

Koedoe ◽

10.4102/koedoe.v57i1.1240 ◽

2015 ◽

Vol 57 (1) ◽

Cited By ~ 14

Author(s):

Morgan B. Pfeiffer ◽

Jan A. Venter ◽

Colleen T. Downs

Keyword(s):

South Africa ◽

Protected Areas ◽

Breeding Season ◽

Movement Ecology ◽

Kernel Density ◽

Eastern Cape Province ◽

Eastern Cape ◽

Convex Polygons ◽

Density Estimates ◽

Kernel Density Estimates

Despite the extent of subsistence farmland in Africa, little is known about endangered species that persist within them. The Cape Vulture (Gyps coprotheres) is regionally endangered in southern Africa and at least 20% of the population breeds in the subsistence farmland area previously known as the Transkei in the Eastern Cape province of South Africa. To understand their movement ecology, adult Cape Vultures (n = 9) were captured and fitted with global positioning system/global system for mobile transmitters. Minimum convex polygons (MCPs),and 99% and 50% kernel density estimates (KDEs) were calculated for the breeding and non breeding seasons of the Cape Vulture. Land use maps were constructed for each 99% KDE and vulture locations were overlaid. During the non-breeding season, ranges were slightly larger(mean [± SE] MCP = 16 887 km2 ± 366 km2) than the breeding season (MCP = 14 707 km2 ± 2155 km2). Breeding and non-breeding season MCPs overlapped by a total of 92%. Kernel density estimates showed seasonal variability. During the breeding season, Cape Vultures used subsistence farmland, natural woodland and protected areas more than expected. In the non-breeding season, vultures used natural woodland and subsistence farmland more than expected, and protected areas less than expected. In both seasons, human-altered landscapes were used less, except for subsistence farmland.Conservation implications: These results highlight the importance of subsistence farm land to the survival of the Cape Vulture. Efforts should be made to minimise potential threats to vultures in the core areas outlined, through outreach programmes and mitigation measures.The conservation buffer of 40 km around Cape Vulture breeding colonies should be increased to 50 km.

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Forest management strategies for dealing with fire-related uncertainty when managing two forest seral stages

Canadian Journal of Forest Research ◽

10.1139/x10-212 ◽

2011 ◽

Vol 41 (2) ◽

pp. 309-320 ◽

Cited By ~ 12

Author(s):

David W. Savage ◽

David L. Martell ◽

B. Mike Wotton

Keyword(s):

Forest Management ◽

Simulation Model ◽

Sustainable Forest Management ◽

Management Strategies ◽

Fire Regimes ◽

Analysis Tool ◽

Planning Model ◽

Minimum Area ◽

Forest Management Planning ◽

Planning Models

Ecological values are an important aspect of sustainable forest management, but little attention has been paid to maintaining these values when using traditional linear programming (LP) forest management planning models in uncertain planning environments. We embedded an LP planning model that specifies when and how much to harvest in a simulation model of a “managed” flammable forest landscape. The simulation model was used to evaluate two strategies for dealing with fire-related uncertainty when managing mature and old forest areas. The two seral stage areas were constrained in the LP planning model to a minimum of 10% of the total forest area and the strategies were evaluated under four representative fire regimes. We also developed a risk analysis tool that can be used by forest managers that wish to incorporate fire-related uncertainty in their decision-making. We found that use of the LP model would reduce the areas of the mature and old forest to their lower bound and fire would further reduce the seral areas below those levels, particularly when the mean annual burn fraction exceeds 0.45% per annum. Increasing the minimum area required (i.e., the right-hand side of the constraint) would increase the likelihood of satisfying the minimum area requirements.

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